number theory - Efficient modular solution to $ax - by equiv 0pmod{p}$?

Given prime $p$, integers $x$ and $y$ where both $x, y < p$ and $x \neq y$, is there an efficient way to find nontrivial coefficients $a, b$ where $a, b < \sqrt p$ such that

$$ax - by \equiv 0\bmod p$$

Further, assume that we are told that such a pair $a, b$ exists; the question is, what's the best way to find them?

If there is no efficient (non-brute force) method, then assume we have many such pairs $a_ix_i - a_jy_j \equiv 0\bmod p$ where the $a_i, a_j$ are known to exist for their respective $x_i, y_j$ pairs, and are bounded by $\sqrt p$ as above; is there an efficient way to find at least one satisfying pair $a_i, a_j$?